{
  "abstract": "In the planted clique problem, one observes either an Erdős--Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = Θ(\\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t \\geq 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k = λ\\sqrt{n}$ planted in a random location with $λ> 1 / \\sqrt{c(t)}$ can be detected and recovered in time $n^{t + 1 + o(1)}$. For instance, since $c(1) = e$, this recovers by counting signed trees the performance of the $\\widetilde{O}(n^2)$-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when $λ> 1 / \\sqrt{e} \\approx 0.6066$. For $t \\geq 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $λ$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{ω+ o(1)}$ for $ω$ the constant of square matrix multiplication and succeeds when $λ> 0.3320$; under the folklore conjecture that $ω= 2$, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques. Second, we show that the above algorithm for $t = 1$ can be combined with the boosting scheme of Alon--Krivelevich--Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $λ$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $λ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24780",
  "author_total": 2,
  "authors": [
    {
      "name": "Dmitriy Kunisky"
    },
    {
      "name": "Songtao Mao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3es3j6tf3pqzn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24780",
  "fulltext_url": null,
  "id": "3es3j6tf3pqzn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24780",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7214044646",
  "pdf_url": "https://arxiv.org/pdf/2609.24780",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3es3j6tf3pqzn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24780",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24780"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Improved polynomial-time algorithms for detecting and recovering planted $Θ(\\sqrt{n})$-cliques",
  "title_text": "Improved polynomial-time algorithms for detecting and recovering planted Θ(√n)-cliques",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/3es3j6tf3pqzn",
  "venue": "arXiv",
  "versions": []
}